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1 Dscusson Paper No. 268 小標本特性に優れたパネル単位根検定 千木良弘朗 山本拓 2011 年 7 月 TOHOKU ECONOMICS RESEARCH GROUP GRADUATE SCHOOL OF ECONOMICS AND MANAGEMENT TOHOKU UNIVERSITY KAWAUCHI, AOBA-KU, SENDAI, JAPAN
2 Λ z Quah (1994) 1 Dckey and Fuller (1979) augmented Dckey-Fuller ADF Levn, Ln and Chu (2002) LLC Quah (1994) ADF Λ Emal: hchgra@econ.tohoku.ac.jp z Emal: yamamoto.taku@nhon-u.ac.jp 1
3 LLC 1 Im, Pesaran and Shn (2003) IPS LLC IPS IPS 1 3 ADF Maddala and Wu (1999) Cho (2001) combnaton ADF combnaton Hurln and Mgnon (2006) IPS Hlouskova and Wagner (2006) IPS LLC ADF combnaton combnaton 2 IPS LLC combnaton LLC IPS ADF combnaton ADF ADF IPS ADF 3 ADF ADF IPS 1 Maddala and Wu (1999) 2 IPS LLC 3 Maddala, Wu andlu (2000) 2
4 2 IPS autoregressve, AR y t = μ + v t ; v t = ff 1 v ;t 1 + ff 2 v ;t ff K v ;t K + " t ; " t ο ::d:n (0;ff 2 ); =1; ;N; t =1; ;T (1) y s = μ + v s ; s = K +1; ; 0 " t (1) y t = μ ff (1) + K X k=1 ff k y ;t k + " t P K ff (1)=1 ff k=1 k ADF y t = ff + f y ;t 1 + K 1 X k=1 ρ k y ;t k + " t (2) ff = μ ff (1) f = ff (1) ρ k = P K h=k+1 ff h 2.1 IPS IPS (2) H 0 : f =0; 8 vs. H 1 : f < 0; =1; ;N 1 ; f =0; = N 1 +1; ;N (3) H 1 IPS 3
5 f < 0 N N 1 lm N!1 N 1 =N = s; 0 <s» " t ::d: IPS T ADF " t movng average, MA ADF AR IPS " t MA AR K MA IPS IPS ADF t c t c (2) 2 ordnary least squares, OLS f t- t c = q ^f Var( ^f ) (4) 4 IPS Maddala and Wu (1999) vs. 5 Taylor andsarno (1998) 4
6 ^f = y0 ; 1M Q y y; 1M 0 ; Q y ; 1 y = (T 1) y ;1 y ;0 y ;2 y ;1. y T y ;T 1 Var( ^f )= ; y ; 1 = (T 1) ^ff 2 " y; 1M 0 ; ^ff 2 " = y0 M X y Q y ; 1 T (K +1) ; y ;0 y ;1. y ;T h Q (T K ) = y ; 1 ::: y ; K +1 ; = 5 ; X (T (K +1)) M X = I T X (X 0 X ) 1 X 0 ; M Q = I T Q (Q 0 Q ) 1 Q 0 = hq y ; 1 ; ; y ; 1; ; y ; K +1 y ; 1 t c T!1 Dckey-Fuller DF DF c " t T H 0 t c ο ::d: c ; =1; ;N t c c Z = p Nf P N =1 t c=n E( c )g p Var( c ) N;T!1; d! N(0; 1) N;T!1; d! T!1 N!1 Z IPS IPS W = p Nf P N =1 t c=n P N =1 E(t c)=ng q PN =1 Var(t c)=n N;T!1; d! N(0; 1) (5) W IPS E(t c ) Var(t c ) K T ρ k " t IPS " t ::d: 6 ρ k = 0 T K E(t c ) Var(t c ) 6 IPS E(t c ) Var(t c ) 5
7 2.2 SCT-IPS IPS ADF (2) suppressed constant term SCT IPS SCT-IPS OLS (2) y ;t 1 E(y ; K +1) =μ + E(v ; K +1) =μ P 1 T y T t=1 ;t 1 y ; K +1 μ SCT (1) v ; K +1 (y t v ; K +1) =μ +(v t v ; K +1); (v t v ; K +1) =ff 1 (v ;t 1 v ; K +1)+ + ff K (v ;t K v ; K +1) ff (1)v ; K +1 + " t y ; K +1 = μ + v ; K +1 (y t y ; K +1) =μ ff (1) + K X k=1 y Λ t = y t y ; K +1 y Λ t = ff + f y Λ ;t 1 + K 1 X k=1 ff k (y ;t k y ; K +1) ff (1)y ; K +1 + " t ρ k y Λ ;t k ff (1)y ; K +1 + " t y Λ t = y t y t = ff + f y Λ ;t 1 + K 1 X k=1 y t = f y Λ ;t 1 + K 1 X k=1 ρ k y ;t k ff (1)y ; K +1 + " t ρ k y ;t k + t ; t = ff ff (1)y ; K +1 + " t (6) 6
8 t 0 IPS (3) (6) OLS ADF (4) t Λ c = q ^f Λ Var( ^f Λ) t Λ c SCT-ADF H 0 SCT-ADF H 0 ff (1) = 0 (6) y t = K 1 X k=1 ρ k y ;t k + " t DF 0 t Λ c T H 0 t Λ c ο ::d: 0 ; =1; ;N t Λ c W Λ = p Nf P N =1 tλ c=n P N =1 E(tΛ c)=ng q PN =1 Var(tΛ c)=n N;T!1; d! N(0; 1) (7) W Λ SCT-IPS H 0 SCT-ADF SCT-IPS t Λ c IPS ρ k =0 data generatng process DGP 1 K =1 K 1 W Λ (7) T N T N =1 DF 0 E( 0 ) Var( 0 ) Abadr (1995) Gonzalo and Ptaraks (1998) Abadr (1995) 0 probablty densty 7
9 1: t 0 K T :347 0:399 0:411 0:415 0: : DGP y t = " t ; " t ο ::d:n (0; 1); t =1; ;T y 0 =0 DGP IPS t 0 K =1 y t = fy t + " t f =0 t- T = 1 Gonzalo and Ptaraks (1998) Nabeya (1999) functon, p.d.f. cumulatve dstrbuton functon, c.d.f. p.d.f. c.d.f. 0 N ( 0:3; 1) 0:3 1 Gonzalo and Ptaraks (1998) E( 0 )= 0:423 Var( 0 )=0:963 0 N ( 0:423; 0:963) Abadr and Lucas (2000) SCT-IPS t Λ c t Λ c Abadr and Lucas (2000) N!1 t Λ c T T 1 t 0 1 t 0 T =10 T = T = 10 ο χ 2 χ T = T = t
10 1: t 0 T =10 T =25 T =50 T =100 T =25 t 0 t 0 N ( 0:399; 1:004 2 ) 0:399 1:004 1 T = 25 t 0 3 SCT-IPS 5% t 0 1:948 N ( 0:399; 1:004 2 ) 2:050 T =25 N =1 SCT-IPS 5% SCT-IPS T 25 N 1 9
11 2: t 0 T : 1 0 T = 1 Nabeya (1999) 3: t 0 qantles(%) t 0 (T =25) 2:660 2:265 1:948 1:611 0: N ( 0:399; 1:004 2 ) 2:735 2:367 2:050 1:686 0: H 1 SCT-ADF H 0 SCT-ADF H 0 y ; K +1 2 H 1 y ; K +1 y t y ; K +1 E(y t y ; K +1) = 0 Var(y t y ; K +1) y ; K +1 Cov((y t y ; K +1)(y ;t+h y ; K +1)) h!1 0 0 K =1 y t = f y Λ ;t 1 + t ; t = ff ff (1)y 0 + " t f = ff 1 1 < 0 OLS ^f Λ Var( ^f Λ) 10
12 : ^f Λ T!1; p! (ff 1 1) + (1 ff 1)(1 ff 2 1)(μ y 0 ) 2 ff 2 +(1 ff2 )(μ 1 y 0 ) 2 T Var( ^f Λ ) T!1; p! (1 ff 2 1)+ 2ff2 (1 ff 2 1)(μ y 0 ) 2 (ff 1 ff 2 1) (1 ff 2 1) 3 (μ y 0 ) 4 (ff 2 +(1 ff 2 1)(μ y 0 ) 2 ) 2 y 0 οn(μ ; ^f Λ ff 2 1 ff 2 1 ) T!1; p! (ff 1 1) + (1 ff 1)χ χ 2 1 T Var( ^f Λ ) T!1; p! (1 ff 2 1)+ 2χ2 1(ff 1 ff 2 1) (1 ff 2 1)χ 2 1 (1 + χ 2 1) 2 : ^f Λ p T ff y 0 οn(μ ; 2 ) μ 1 ff 2 ff 2 1 0» (μ y 0 ) 2 < 1 ff 1 1» plm^f Λ < 0 ff T Var( ^f Λ ) y 0 μ ff 2 0» (μ y 0 ) 2 < 1 0 < plmt Var( ^f Λ )» 1 ff 2 1 SCT-ADF (μ y 0 ) 2 plm^f Λ ff 1 1 plmt Var( ^f Λ ) 1 ff 2 1 (μ y 0 ) 2 =0 plm^f Λ = ff 1 1 plmt Var( ^f Λ )=1 ff 2 1 SCT H 1 f 0 11
13 4: N test T sze power sze power sze power sze power 1 IPS SCT-IPS IPS SCT-IPS IPS SCT-IPS IPS SCT-IPS IPS SCT-IPS : SCT-IPS (7) t Λ c T 1 IPS (5) t c T IPS Table 3 DGP y t =(1 ff 1 )μ + ff 1 y ;t 1 + " t ; " t ο ::d:n (0;ff 2 ); = 1; ;N; t = 51; 50; ;T μ N (0; 1) ff 2 U [0:5; 1:5] ff K = SCT-IPS DGP y t =(1 ff 1 )μ +ff 1 y ;t 1 +" t ; " t ο ::d:n (0;ff 2 ); =1; ;N; t = 51; 50; ;T μ N (0; 1) ff 2 U[0:5; 1:5] ff N = f1; 10; 25; 50; 100g T = f10; 25; 50; 100g K =1 IPS SCT-IPS (7) t Λ c T 1 IPS (5) t c T IPS Table SCT-IPS N 12
14 2.2.1 IPS IPS N =1 DF c Nabeya (1999) c N =1 IPS SCT-IPS T =10 N =1 IPS IPS T N T =10 N =50 SCT-IPS IPS SCT-IPS IPS SCT-IPS T ff 1 IPS N =1 ff 1 = f0; 0:3; 0:5; 07; 0:9; 0:95g T =10 ff 1 SCT-IPS IPS T =25 ff 1 = f0; 0:3; 0:5g IPS T =50 T =100 ff 1 = f0; 0:3; 0:5; 0:7g IPS OLS ^f SCT ^f Λ mean squared error, MSE ^f ^f Λ ^f Λ T ff 1 1 MSE ^f MSE Tanaka (1983) Yamamoto and Kuntomo (1984) ^f Λ MSE SCT T!1 ^f ^f Λ T SCT T!1 SCT-IPS 13
15 N 5: SCT-IPS T sze power sze power sze power sze power : SCT-IPS (7) t Λ c T 1 T = 1 4 T SCT-IPS t Λ c Tanaka (1983) AR t- SCT SCT t Λ c SCT-IPS (7) T 1 T T T = 1 SCT-IPS 5 T =50 N T T = 100 T 100 T 4 SCT SCT-IPS IPS 14
16 SCT-IPS N T N =1 T =25 SCT-IPS IPS Ellott, Rothenberg and Stock (1996) SCT ADF Ellott, Rothenberg and Stock (1996) ADF Cho (2001) IPS 3.9 Cho (2001) Cho (2001) 2 SCT 3 Abadr, K.M. (1995): The Lmtng Dstrbuton of the t Rato under a Unt Root," Econometrc Theory, 11, Abadr, K.M. and A. Lucas (2000): Quantles for t-statstcs Based on M-Estmators of Unt Roots," Economcs Letters, 67, Cho, I. (2001): Unt Root Tests for Panel Data," Journal of Internatonal Money and Fnance, 20, Dckey, D.A., and W.A. Fuller (1979): Dstrbuton of the Estmators for Autoregressve Tme Seres wth a Unt Root," Journal of the Amercan Statstcal Assocaton, 74,
17 Ellott, G., Rothenberg, T.J., J.H. Stock (1996): Effcent Tests for an Autoregressve Unt Root," Econometrca, 64, Gonzalo, J. and J.-Y. Ptaraks (1998): On the Exact Moments of Asymptotc Dstrbutons n an Unstable AR(1) wth Dependent Errors," Internatonal Economc Revew, 39, Hlouskova, J. and M. Wagner (2006): The Performance of Panel Unt Root and Statonarty Tests: Results from a Large Scale Smulaton Study," Econometrc Revews, 25, Hurln, C. and V. Mgnon (2006): Second Generaton Panel Unt Root Tests," mmeo. Im, K.S., Pesaran, M.H. and Y. Shn (2003): Testng for Unt Roots n Heterogeneous Panels," Journal of Econometrcs, 115, Levn, A., Ln, C.F. and J. Chu (2002): Unt Root Tests n Panel Data: Asymptotc and Fnte-Sample Propertes," Journal of Econometrcs, 108, Maddala, G.S. and S. Wu (1999): A Comparatve Study of Unt Root Tests wth Panel Data and a New Smple Test," Oxford Bulletn of Economcs and Statstcs, 61, Maddala, G.S., Wu, S. and C. Lu (2000): Do Panel Data Rescue the Purchasng Power Party (PPP) Theory?" n Krshnakumar, J. and E. Ronchett eds. Panel Data Econometrcs: Future Drectons: Papers n Honour of Professor Petro Balestra, Elsever, Nabeya, S. (1999): Asymptotc Moments of Some Unt Root Test Statstcs n the Null Case," Econometrc Theory, 15, Quah, D. (1994): Explotng Cross-Secton Varaton for Unt Root Inference n Dynamc Data," Economcs Letters, 44, Tanaka, K. (1983): Asymptotc Expansons Assocated wth the AR(1) Model wth Unknown Mean," Econometrca, 51,
18 Taylor, M.P. and L. Sarno (1998): The Behavor of Real Exchange Rates Durng the Post-Bretton Woods Perod," Journal of Internatonal Economcs, 46, Toda, H.Y. and T. Yamamoto (1995): Statstcal Inference n Vector Autoregressons wth Possbly Integrated Processes," Journal of Econometrcs, 66, Yamamoto, T. and N. Kuntomo (1984): Asymptotc Bas of the Least-Squares Estmator for Multvarate Autoregressve Models," Annals of the Insttute of Statstcal Mathematcs, 36,
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